Proposition 16.4.6 (Maximal subgroupoids). Passing to maximal subgroupoids defines a canonical lax symmetric monoidal functor \[ (-)^{\simeq }\colon \CMon (\Cat _{\infty })\longrightarrow \CMon (\An ), \] where both categories carry the symmetric monoidal structures of Proposition 16.4.1.
Proof. The inclusion \(\An \hookrightarrow \Cat _{\infty }\) is a symmetric monoidal left adjoint for the cartesian monoidal structures, preserves finite products, and has right adjoint \((-)^{\simeq }\). The result is therefore the commutative-monoid instance of Proposition 16.4.2. □
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